Selecting a material for use as the expansive element

ABSTRACT

The invention concerns thermoelastic designs incorporating and expansive element formed from material selected in accordance a procedure involving the derivation of an indicator of the material&#39;s potential effectiveness for each application.

[0001] Continuation application of U.S. Ser. No. 09/693,079 filed onOct. 20, 2000

CO-PENDING APPLICATIONS

[0002] Various methods, systems and apparatus relating to the presentinvention are disclosed in the following co-pending applications filedby the applicant or assignee of the present invention on May 23, 2000:09/575,197, 09/575,195, 09/575,159, 09/575,132, 09/575,123, 09/575,148,09/575,130, 09/575,165, 09/575,153, 09/575,118, 09/575,131, 09/575,116,09/575,144, 09/575,139, 09/575,186, 09/575,185, 09/575,191, 09/575,145,09/575,192, 09/575,181, 09/575,193, 09/575,156, 09/575,183, 09/575,160,09/575,150, 09/575,169, 09/575,184, 09/575,128, 09/575,180, 09/575,149,09/575,179, 09/575,187, 09/575,155 09/575,133, 09/575,143, 09/575,196,09/575,198 09/575,178, 09/575,164, 09/575,146, 09/575,174, 09/575,163,09/575,168, 09/575,154, 09/575,129 09/575,124, 09/575,188, 09/575,189,09/575,162, 09/575,172, 09/575,170, 09/575,171, 09/575,161, 09/575,141,09/575,125, 09/575,142, 09/575,140, 09/575,190, 09/575,138, 09/575,126,09/575,127, 09/575,158, 09/575,117, 09/575,147, 09/575,152, 09/575,176,09/575,151 09/575,177, 09/575,175 09/575,115, 09/575,114, 09/575,113,09/575,112, 09/575,111, 09/575,108, 09/575,109, 09/575,110, 09/575,182,09/575,173, 09/575,194, 09/575,136, 09/575,119, 09/575,135, 09/575,157,09/575,166, 09/575,134, 09/575,121, 09/575,137, 09/575,167, 09/575,120,09/575,122

[0003] The disclosures of these co-pending applications are incorporatedherein by cross-reference.

FIELD OF THE INVENTION

[0004] The present invention relates to materials potentially suitablefor use as the expansive element in thermoelastic design and to methodsfor ranking the potential relative suitabilities of those materials.

[0005] The invention as developed originally as a means of identifyingand ranking a range of materials that potentially may exhibit superiorproperties for use in the manufacture of microscopic thermal bendactuators for use in micro-electro mechanical systems (MEMS), and willbe described hereinafter with reference to this field. However, it willbe appreciated that the invention is not limited to this particular useand is equally applicable to macroscopic design even though the overalldesign considerations are vastly different and certainly less complex.

BACKGROUND OF THE INVENTION

[0006] It is important to clarify that thermoelastic actuation ischaracterized using force, deflection and temperature as opposed toswitching, which is characterized using deflection and temperature risealone. Macroscopic thermoelastic actuators are typically used asswitches that activate other more energy efficient actuation systems,however, microscopic thermoelastic actuators are an attractive actuationmechanism for a number of reasons. This includes the down scaling ofcertain physical phenomena. For example, it is possible to fabricatevery thin films that decrease the thermal mass and minimize efficiencylosses. Opposing gravitational and inertial forces become negligible onthe microscopic scale. Other advantages include ease of fabrication(although more complex than simple electrostatic actuators) and thepossibility of low voltage operation. Disadvantages include a lowoperational bandwidth determined by the thermal conductivities ofsubstrate materials—this is more of an advantage for the currentapplication allowing for rapid firing.

[0007] A relatively diverse range of output force and deflection valuescan be obtained by altering actuator geometry. However, the fundamentaloperation of actuation is directly related to the mechanical and thermalproperties of the component materials. Correct material selection inassociation with effective design can result in either a smaller or amore efficient actuator. Such an actuator increases wafer yield and isthus more commercially viable. A more efficient actuator may be batterypowered increasing operation simplicity and negating the requirement forexpensive voltage transformers. An increase in thermal efficiencyimproves the operational firing frequency, and decreases the possibilityof thermal crosstalk. This is especially relevant for arrays of thermalactuators in a micro-cilia device.

[0008] However, material selection for MEMS application is notstraightforward. Firstly, published thin film properties can varygreatly due to different fabrication methods and difficulties associatedwith experimentally quantifying material properties on the microscopicscale. Secondly, certain thin films can only be fabricated with certainlayer thicknesses because inherent stress can shatter or curl thesubstrate wafer. Thirdly, only certain materials can be used in thefabrication process at most fabs as the introduction of a new materialcan contaminate machinery.

[0009] Progress to Date

[0010] Until recently, the only materials regularly used or consideredfor use in such applications were polysilicon, single crystal silicon.However, the applicant just previously made the surprising discoverythat titanium nitride and titanium boride/diboride exhibited excellentproperties relevant to this application.

[0011] Realizing the breakthrough this surprising discovery signified,the applicant sought to try and identify possible alternatives in orderto provide designers of thermoelastic systems with more choice andflexibility. However, given the lack of available data on their filmproperties for various materials and the fact that empirical testingwith MEMS would be prohibitively expensive, there was clearly a need, orit was at least highly desirable to be able to determine a method ofevaluating materials for this use based solely on the commonly availablemacro material properties.

SUMMARY OF THE INVENTION

[0012] It is therefore an ultimate object of one aspect of thisinvention to identify a range of alternative materials that willpotentially exhibit superior properties for use in thermoelastic designand of another aspect to provide a means of ranking the potentialsuitability of a given range of materials for this same use.

[0013] According to a first aspect of the invention there is provided amethod of selecting a material for use as the expansive element in athermoelastic design by deriving an indicator of the material'spotential effectiveness for that use, said method including the step ofcalculating a dimensionless constant εγ for that material in accordancewith the formula:${ɛ\quad \gamma} = \frac{E\quad \gamma^{2}T}{\rho \quad C}$

[0014] wherein E is the Young's modulus of the material; γ is thecoefficient of thermal expansion; T is the maximum operatingtemperature, ρ is the density and C is the specific heat capacity.

[0015] In accordance with a second aspect the invention, in anotherbroad form, also provides a method of manufacturing a thermoelasticelement that includes at least one expansive element, the methodincluding:

[0016] selecting a material for use as the expansive element in thethermoelastic design by deriving an indicator of the material'spotential effectiveness for that use, said method including the step ofcalculating a dimensionless constant εγ for that material in accordancewith the formula:${ɛ\quad \gamma} = \frac{E\quad \gamma^{2}T}{\rho \quad C}$

[0017] wherein E is the Young's modulus of the material; γ is thecoefficient of thermal expansion; T is the maximum operatingtemperature, ρ is the density and C is the specific heat capacity andselecting the material on the basis of ε, and

[0018] manufacturing the thermoelastic element with the at least oneexpansive element formed of the selected material.

[0019] Preferably, the method of selection includes the step ofnormalizing the dimensionless constant relative to that of silicon to avalue ε which is achieved by deriving the value εγ for the material ofinterest at the relevant temperature value and dividing this by thevalue of ε obtained for silicon at that same temperature.

[0020] The relevant maximum operating temperature will depend upon thesurrounding materials and their function but is most commonly theoxidizing temperature or the melting point temperature.

[0021] Desirably, the selection method includes the step of eliminatingcertain materials by requiring a pre-determined resistivity range. Inone preferred form this resistivity range is between 0.1 μΩm and 10.0μΩm.

[0022] In accordance with a third aspect of the invention there isprovided an expansive element in a thermoelastic design that is madefrom any functionally suitable material or combinations of materialsselected from a group including:

[0023] silicides and carbides of titanium.

[0024] In accordance with a fourth aspect of the invention there isprovided an expansive element in a thermoelastic design that is madefrom any functionally suitable material or combinations of materialsselected from a group including:

[0025] borides, silicides, carbides and nitrides of tantalum,molybdenum, niobium, chromium, tungsten, vanadium, and zirconium.

[0026] In accordance with a fifth aspect of the invention there isprovided an expansive element in a thermoelastic design that is madefrom any functionally suitable alloy material or combinations of alloymaterials selected from the group including:

[0027] borides, silicides, carbides and nitrides of titanium, tantalum,molybdenum, niobium, chromium, tungsten, vanadium, and zirconium.

[0028] Preferably the expansive element in a thermoelastic design inaccordance with the third, fourth or fifth aspect of the invention alsoincludes one or more of the following properties:

[0029] (a) a resistivity between 0.1 μΩm and 10.0 μΩm;

[0030] (b) chemically inert in air;

[0031] (c) chemically inert in the chosen ink; and

[0032] (d) depositable by CVD, sputtering or other thin film depositiontechnique.

BRIEF DESCRIPTION OF THE DRAWINGS

[0033] Derivation of the dimensionless constant ε of the first aspect ofthe invention, together with sample applications and examples of derivedvalues of this constant and other properties for a range of materials,will now be described in detail with reference to the accompanyingdrawings in which:

[0034]FIG. 1 shows a schematic representation of a thermoelasticactuator;

[0035]FIG. 2 shows a plot of longitudinal work versus heat energy forsingle material clamped/free titanium beam (length 20 μm, thickness 1μm, width 5 μm);

[0036]FIG. 3 shows a plot derived from FIG. 2 of expansion efficiencyversus temperature efficiency for a clamped/free titanium beam; and

[0037]FIG. 4 shows a comparison of mechanical work versus the heatenergy of thermoelastic actuator fabricated from Titanium and Silicon.

DETAILED DESCRIPTION

[0038] A non-dimensionalized material actuation efficiency is presentedthat assesses the potential application of a material to thermoelasticdesign. The method is based on the material thermal and mechanicalproperties and assists in a structured approach of material selectionfor effective design.

[0039] The Material Actuation Efficiency

[0040] Actuators are characterized by a combination of deflection, forceand operation temperature in contrast to switches that are characterizedby operation temperature and deflection alone. Fundamental thermoelasticdesign is characterized by the differential longitudinal expansion oftwo bonded layers. Thus, the expansion of isolated unbonded layersdirectly relates to global behavior. A single material beam is used hereto illustrate the material actuation efficiency. The approach isstraightforward and relates to general thermoelastic design. Thederivation assumes that material properties are constant across thethermal range.

[0041] Equations 1 to 3 are fundamental thermomechanical equationsdescribing the behavior of simple single material beam subjected to aquantity of heat, Q as illustrated in FIG. 1. Equation 1 describes theextension, δL, of a free/free beam and equation 2 describes the reactionforce, F, of a clamped/clamped beam.

δL=γL ₀ T  (EQ 1)

[0042] Where: δL=extension of beam, L₀=original length of beam,T=operation temperature (temperature rise), and γ=coefficient of thermalexpansion of beam.

F=AEγT  (EQ2)

[0043] F=force exerted by beam expansion, A=cross sectional area ofbeam, E=Young's Modulus.

Q=VρCt  (EQ3)

[0044] Where: Q=heat energy input, V=volume of beam, ρ=density, andC=specific heat capacity of beam.

[0045] Potential mechanical work is given by equation 4 and is definedas the product of the clamped beam force, F, and free beam deflection,δL. The quadratic relationship between the heat input and outputmechanical work for the simple monolithic beam is shown in FIG. 18.

W=FδL  (EQ4)

[0046] Where: W=mechanical work

[0047] Equation 5 describes the non-dimensional thermoelastic actuationefficiency and is formulated as the quotient of the mechanical work andheat energy as described by equations 3 and 4. The efficiency isindependent of geometry and is a primary indication of a material'spotential application to thermoelastic design. The linear relationshipbetween the actuation efficiency and material temperature for the simplebeam is shown in FIG. 3. The graph indicates that high temperatureoperation is desirable for maximum efficiency. The plot is limited bythe applicable operation temperature and therefore, different materialplots are of different lengths. The assumption used in this text is thatthe operation temperature is the material melting point because it isindicative of the operable thermal range. Thus, the material actuationefficiency, ε, is defined as the actuation efficiency at the maximumoperable temperature, T, of that material. The slope of the efficiencycurve is a constant, m_(ε) and is defined in equation 6. The combinationof ε and m_(ε) fully characterize a materials actuation characteristicsnon graphically. $\begin{matrix}\begin{matrix}{ɛ = \frac{{Output}\quad {Mechanical}\quad {Work}}{{Heat}\quad {Energy}\quad {Input}}} \\{= {\frac{E\quad \gamma^{2}T}{\rho \quad C}\left\lbrack \frac{\left( {N\text{/}m^{2}} \right)\left( {1\text{/}{{{^\circ}C}.^{2}}} \right)\left( {{{^\circ}C}.} \right)}{\left( {{kg}\text{/}m^{3}} \right)\left( {{Nm}\text{/}{kg}\quad {{{^\circ}C}.}} \right)} \right\rbrack}}\end{matrix} & \left( {{EQ}\quad 5} \right) \\{m_{ɛ} = {\frac{ɛ}{T} = {\frac{E\quad \gamma^{2}}{\rho \quad C}\left\lbrack \frac{N\text{/}m^{2}\quad 1\text{/}{{^\circ}C}^{2}}{{kg}\text{/}m^{3}\quad {Nm}\text{/}{kg}\quad {{{^\circ}C}.}} \right\rbrack}}} & \left( {{EQ}\quad 6} \right)\end{matrix}$

[0048] Material Selection

[0049] Different thin film materials including materials with extremeproperties (PTFE—high g, Diamond—high E) and compounds from all themajor CVD groups including borides, silicides, nitrides and carbides isshown in Table 2. The efficiency values are scaled according to siliconefficiency values because the inclusion of scaled values greatlysimplifies design equations described in the following text. The scalingor comparison of a material with respect to a reference material is anintegral step in the material selection process. In addition, scalingalso results in a more readable index as illustrated by the followingcomparisons. Silicon is chosen as the reference material because of itspredominance in lithographic fabrication.

[0050] Preliminary candidates for thermoelastic actuation can beselected according to efficiencies and slopes, however, it is importantto note that two materials that have identical ε but differing m_(ε)will output different amounts of work for any constant geometry (seeComparison 1 below, different amounts of heat energy are also required).Three important design parameters are defined here as heat input, workoutput and volume. A design matrix can be constructed by varying eachparameter and can then be used to select suitable materials. Thefollowing comparisons are used to assemble the design matrix. TABLE 2Material Properties. g E r C m_(e)/m_(r,e) O.T M.P. MN MN KXX R Material10⁻⁶/° C. GPa kg/m³ J/kg ° C. ° C.⁻¹ ° C. ° C. O.T. M.P. W/m.K mWmAluminum 23.1 68.9 2700 897 17.12 657 7.98 231 0.027 Boron Carbide 4.5454 2520 955 4.31 2450 7.49 35 5e4 Chromium diBoride 11.1 540 5600 69019.42 1000 2150 13.78 29.62 32 0.18 Chromium diSilicide 5.9 5600 11501560 0.8 Chromium Carbide 9.9 385 6680 530 12.02 1100 1895 9.38 16.16 190.75 Chromium Oxide 9.0 102 5210 730 2.45 1000 2603 1.74 4.52 30 13Copper 16.5 110 8940 386 9.79 1085 7.53 398 0.017 Gold 14.2 80 19300 1297.31 1064 5.52 315 0.023 Hafnium Carbide 6.3 410 12670 190 7.63 600 39303.24 21.25 13 0.4-0.6 Hafnium diBoride 7.6 11200 300 1500 3250 51 0.1Hafnium diSilicide 8030 1100 1700 Hafnium Monocarbide 6.5 424 11940 38908 0.5 Hafnium Nitride 6.5 13,940 500 3300 17 32 Molybdenum 4.8 343 10200251 3.48 2623 6.48 138 Molybdenum Boride 5 685 7480 530 4.87 1000 21403.46 7.40 27 0.18 Molybdenum Carbide 6.7 530 9120 315 9.34 500 2500 3.3116.56 22 0.57 Molybdenum diSilicide 8.4 450 6240 550 10.44 1700 205012.58 15.17 49 0.7 Nickel 13.4 200 8900 444 10.25 1455 10.58 90.7Niobium diBoride 8.6 650 7210 420 17.91 850 3000 10.80 38.10 0.12 17Niobium diSilicide 8.5 5690 900 2050 0.5 Niobium Carbide 7.4 450 7820290 12.26 650 3500 5.65 30.42 14 0.19 PTFE 220 1.3 2130 1024 32.54 2004.62 140 10e22 Silicon 3.0 162 2330 705 1.00 1410 1410 1 1 149 2300Silicon Carbide 4.7 304 3440 669 3.29 2700 6.30 90 0.5 Tantalum Carbide6.7 510 14500 190 9.37 650 3900 4.32 25.93 23 0.35 Tantalum diBoride 8.5250 12600 250 6.47 850 3090 3.90 14.17 16 0.14 Tantalum diSilicide 9.59080 360 800 2670 0.46 Titanium Carbide 7.4 462 4920 480 12.08 700 31606.00 27.08 17.2 1.55 Titanium difloride 8.2 575 4450 632 15.51 1400 325315.40 35.78 26.4 0.13 Titanium diSilicide 10.7 270 4100 480 17.72 13001540 16.34 19.35 46 0.145 Titanium Nitride 9.4 600 5450 636 17.25 5002950 6.12 36.10 30 1.35 Tungsten Boride 5.0 790 13100 460 3.70 1000 23652.62 6.20 52 0.19 Tungsten Carbide 5.2 690 15800 200 6.66 500 2780 2.3613.13 29 0.2 Tungsten diSilicide 7.0 300 9750 330 5.15 1200 2165 4.397.91 48 33e10 Vanadium diBoride 7.6 260 5100 670 4.96 600 2430 2.11 8.5442 0.13 Vanadium Carbide 6.7 420 5480 530 7.32 600 2730 3.12 14.18 100.59 Vanadium diSilicide 11.2 5100 1000 1700 25 0.66 Vanadium Nitride8.1 460 6080 630 8.89 450 2170 2.84 13.68 5.2 0.85 Zirconium Carbide 6.3410 6560 250 11.19 600 3440 4.76 27.31 22 0.42 Zirconium diBoride 5.9340 6170 1300 3245 58 0.15 Zirconium diSilicide 8.7 270 4900 1150 160015 0.76 Zirconium Nitride 5.9 500 7350 400 6.68 500 2950 2.37 13.97 100.2-0.3

[0051] Comparison 1

[0052] The mechanical work and heat input between a material and siliconfor a constant beam volume is compared. Thus, Comparison 1 calculatesthe maximum possible relative work and associated relative heat inputrequired due to a direct material substitution. Details of thecomparison for different materials are included in Table 3 which showsthat CVD ceramics are far superior actuator materials than silicon(Table 3 is formulated using melting point and Table 4 is formulatedusing oxidation temperature). Titanium nitride can output 159.3 timesmore the amount of mechanical work than silicon with only 4.41 times theamount of heat input. The factor in equation 8 and the scaled materialefficiency ratio (as included in Table 2) repeatedly occur in thefollowing comparisons illustrating the versatility of the method.$\begin{matrix}{\frac{W_{c}}{W_{r}} = {\frac{ɛ_{c}Q_{c}}{ɛ_{r}Q_{r}} = {\frac{ɛ_{c}}{ɛ_{r}}\left( \frac{\rho_{c}C_{c}T_{c}}{\rho_{r}C_{r}T_{r}} \right)}}} & \left( {{EQ}\quad 7} \right)\end{matrix}$

[0053] The r subscript denotes the reference material which is siliconin this case. The c subscript denotes the compared material.$\begin{matrix}{\frac{Q_{c}}{Q_{r}} = \left( \frac{\rho_{c}C_{c}T_{c}}{\rho_{r}C_{r}T_{r}} \right)} & \left( {{EQ}\quad 8} \right)\end{matrix}$

[0054] Comparison 2

[0055] Different materials increase in temperature by different amountswhen subjected to the same quantity of heat energy for a constantvolume. The material volume is scaled relative to the silicon volumeaccording to the constraints that the same amount of silicon heat energyis input to both actuators and the compared material attains itsoperational temperature. Thus, the actuation efficiency value remainsunchanged because it is not a function of volume and the operabletemperature is reached (as equation 5 shows). Comparison 2 representsthe design case where heat and volume are critical factors.

[0056] The scaled volume and output mechanical work are calculated usingequations 9 and 10. The volume change is typically implemented bymodifying one geometric dimension, i.e. length, width or thickness.Titanium nitride is capable of 36.1 times the amount of work thatsilicon is capable with the same heat energy input but with only 0.23times the volume. Equation 9 is the inverse of equation 8 and equation10 is simply the scaled efficiency number as included in Table 2.$\begin{matrix}{Q_{r} = {{V_{r}\rho_{r}C_{r}T_{r}} = {Q_{c} = {\left. {V_{c}\rho_{c}C_{c}T_{c}}\Rightarrow\frac{V_{({c,{Qr}})}}{V_{r}} \right. = \frac{\rho_{r}C_{r}T_{r}}{\rho_{c}C_{c}T_{c}}}}}} & \left( {{EQ}\quad 9} \right)\end{matrix}$

[0057] The first entry of the bracketed subscript in these equationsrefers to the material that the beam is constructed from. The secondentry refers to the constraining variable for the described parameter.For example—W_((c,Vc))=Mechanical work output from beam constructed ofcompared material with a volume of V_(c). $\begin{matrix}{\frac{W_{({c,{Vc}})}}{W_{({r,{Vr}})}} = {\frac{ɛ_{c}Q_{r}}{ɛ_{r}Q_{r}} = \frac{ɛ_{c}}{ɛ_{r}}}} & \left( {{EQ}\quad 10} \right)\end{matrix}$

[0058] Comparison 3

[0059] The output mechanical work resulting from silicon heat energy forconstant volume beams is compared. The operation temperature andefficiency value for the compared material changes. However, the newefficiency is easily calculated using a multiplicative ratio of the newand old operation temperatures because of the linear relationshipbetween temperature and efficiency (as shown in FIG. 3). The newoperation temperature and work are given by equations 11 and 12. Thiscomparison represents the design case where heat is a criticalparameter.

[0060] PTFE will melt when subjected to the input silicon heat value.Titanium disilicide outperforms titanium nitride mainly because of thehigher computed operating temperature (Table 3). $\begin{matrix}{Q_{r} = {{V_{r}\rho_{r}C_{r}T_{r}} = {Q_{c} = {\left. {V_{c}\rho_{c}C_{c}T_{({c,{Qr}})}}\Rightarrow T_{({c,{Qr}})} \right. = {T_{r}\left( \frac{\rho_{r}C_{r}}{\rho_{c}C_{c}} \right)}}}}} & \left( {{EQ}\quad 11} \right)\end{matrix}$

[0061] Comparison 4 $\begin{matrix}{\frac{W_{({c,{Qr}})}}{W_{({r,{Qr}})}} = {\frac{ɛ_{({c,{Qr}})}Q_{r}}{ɛ_{r}Q_{r}} = {\frac{T_{({c,{Qr}})}ɛ_{2}}{T_{c}ɛ_{r}} = {\left( \frac{\rho_{r}C_{r}T_{r}}{\rho_{c}C_{c}T_{c}} \right)\frac{ɛ_{c}}{ɛ_{r}}}}}} & \left( {{EQ}\quad 12} \right)\end{matrix}$

[0062] The material volume is scaled with respect to the silicon volumeaccording to the constraint that the compared material operationtemperature and silicon work are maintained. Thus, if the silicon workvalue is less then the original work then the volume is scaled down.Otherwise the volume is increased as is the case for PTFE or amorphousSilicon Dioxide. The material actuation efficiency reoccurs in thecalculations as an inverse as shown in equation 14

[0063] Titanium nitride can output the same amount of work as siliconbut with a volume that is less than two orders of magnitude smaller withan input heat energy that is less than an order smaller. $\begin{matrix}{W_{r} = {{V_{r}E_{r}\gamma_{r}^{2}T_{r}^{2}} = {W_{c} = {\left. {V_{c}E_{c}\gamma_{c}^{2}T_{c}^{2}}\Rightarrow\frac{V_{({c,{Wr}})}}{V_{r}} \right. = \frac{E_{r}\gamma_{r}^{2}T_{r}^{2}}{E_{c}\gamma_{c}^{2}T_{c}^{2}}}}}} & \left( {{EQ}\quad 13} \right) \\{\frac{Q_{({c,{Vc}})}}{Q_{({r,{Vr}})}} = {\frac{ɛ_{r}W_{r}}{ɛ_{c}W_{r}} = \frac{ɛ_{r}}{ɛ_{c}}}} & \left( {{EQ}\quad 14} \right)\end{matrix}$

[0064] Comparison 5

[0065] The input heat energy required to output silicon mechanical workfor constant volume beams is compared. The operation temperature andthus efficiency value for the compared material changes. The newefficiency can be calculated in an identical fashion to that describedin comparison 3. The operational temperature and heat input values arecalculated using equations 15 and 16.

[0066] The table shows that titanium disilicide slightly outperformstitanium nitride whereas both PTFE and silicon dioxide will melt. TheCVD ceramics are again shown to have the best performance.$\begin{matrix}{W_{r} = {{V_{r}E_{r}\gamma_{r}^{2}T_{r}^{2}} = {W_{c} = {\left. {V_{c}E_{c}\gamma_{c}^{2}T_{c}^{2}}\Rightarrow T_{({c,{Wr}})} \right. = {\left( \frac{\gamma_{r}}{\gamma_{c}} \right)\sqrt{\frac{E_{r}}{E_{c}}}}}}}} & \left( {{EQ}\quad 15} \right) \\{\frac{Q_{({c,{Wr}})}}{Q_{({r,{Wr}})}} = {\frac{ɛ_{r}W_{r}}{ɛ_{({c,{Qr}})}W_{r}} = {\frac{ɛ_{r}T_{c}}{ɛ_{c}T_{({c,{Qr}})}} = {\frac{ɛ_{r}T_{c}\gamma_{c}}{ɛ_{c}T_{r}\gamma_{r}}\sqrt{\frac{E_{c}}{E_{r}}}}}}} & \left( {{EQ}\quad 16} \right)\end{matrix}$

TABLE 3 Design comparisons for materials included in Table 2.Comparisons are done using melting point temperature Comparison 1Comparison 2 Comparison 3 Comparison 4 Comparison 5 Constant V Q V,Q WV,W V_((c,Qr))/ W_((c,Vc))/ W_((c,Wr))/ V_((c,Wr))/ Q_((c,Vc))/Q_((c,Wr))/ Q_(c)/Q_(r) W_(c)/W_(r) V_((r,Qr)) W_((r,Vr)) T_((c,Qr))W_((r,Qr)) V_((r,Vr)) Q_((r,Vr)) T_((c,Wr)) Q_((r,Wr)) Aluminum 0.695.48 1.46 7.98 >Tmelt 0.183 0.125 280.79 0.29 Boron Carbide 2.55 19.060.39 7.49 962.41 2.94 0.053 0.133 561.51 0.58 Chromium diBoride 3.59106.23 0.28 29.62 599.41 8.26 0.009 0.0330 208.73 0.35 Chromium Carbide2.90 46.80 0.35 16.16 654.20 5.58 0.021 0.062 277.16 0.42 Chromium Oxide4.27 19.34 0.23 4.52 608.98 1.06 0.052 0.221 592.32 0.97 Copper 1.6212.18 0.62 7.53 671.18 4.66 0.082 0.132 311.11 0.46 Gold 1.14 6.31 0.875.52 930.29 4.82 0.159 0.181 423.90 0.46 Hafnium Carbide 4.08 86.81 0.2421.25 962.13 5.20 0.012 0.047 422.05 0.44 Molybdenum 2.90 18.78 0.346.48 904.67 2.23 0.053 0.154 605.63 0.67 Molybdenum Boride 3.66 27.090.27 7.40 584.23 2.02 0.037 0.135 411.42 0.70 Molybdenum Carbide 3.1051.36 0.32 16.56 806.23 5.34 0.019 0.061 349.05 0.43 MolybdenumdiSilicide 3.04 46.09 0.33 15.17 674.86 4.99 0.022 0.066 302.14 0.45Nickel 2.48 26.26 0.40 10.58 586.13 4.26 0.038 0.095 284.10 0.48 NiobiumdiBoride 3.92 149.44 0.25 38.10 764.86 9.71 0.007 0.026 245.55 0.32Niobium Carbide 3.43 104.26 0.29 30.42 1021.31 8.88 0.010 0.032 342.970.34 PTFE 0.19 0.87 5.31 4.62 >Tmelt 1.152 0.216 >Tmelt Silicon 1.001.00 1.00 1 1410.00 1.00 1.000 1 1410.00 1.00 Silicon Carbide 2.68 16.910.37 6.30 1006.42 2.35 0.059 0.158 657.00 0.65 Tantalum Carbide 4.64120.27 0.22 25.93 840.70 5.59 0.008 0.038 355.83 0.42 Tantalum diBoride4.20 59.57 0.24 14.17 735.28 3.37 0.017 0.071 400.60 0.54 Titanium 1.707.27 0.59 4.28 984.12 2.52 0.138 0.234 619.87 0.63 Titanium diBoride3.95 141.32 0.25 35.78 823.54 9.06 0.007 0.028 273.81 0.33 TitaniumdiSilicide 1.31 25.32 0.76 19.35 1176.90 14.79 0.040 0.0517 306.22 0.26Titanium Nitride 4.41 159.36 0.23 36.10 668.21 8.18 0.006 0.0277 233.830.35 Tungsten Boride 6.15 38.16 0.16 6.20 384.36 1.01 0.026 0.161 383.101.00 Tungsten Carbide 3.79 49.80 0.26 13.13 732.95 3.46 0.020 0.076394.10 0.54 Tungsten diSilicide 3.01 23.80 0.33 7.91 719.86 2.63 0.0420.126 444.06 0.62 Vanadium diBoride 3.58 30.63 0.28 8.54 677.83 2.380.033 0.117 439.34 0.65 Vanadium Carbide 3.42 48.53 0.29 14.18 797.464.14 0.021 0.071 392.10 0.49 Vanadium Nitride 3.59 49.09 0.28 13.68604.67 3.81 0.020 0.0731 309.91 0.51 Zirconium Carbide 2.44 66.51 0.4127.31 1412.28 11.21 0.015 0.0366 422.05 0.30 Zirconium Nitride 3.7452.32 0.27 13.97 787.80 3.73 0.019 0.0716 408.09 0.52

[0067] TABLE 4 Design comparisons for material included in Table 2.Comparisons are done using oxidation temperature Comparison 1 Comparison2 Comparison 3 Comparison 4 Comparison 5 Constant V Q V,Q W V,WV_((c,Qr))/ W_((c,Vc))/ W_((c,Qr))/ V_((c,Wr))/ Q_((c,Vc))/ Q_((c,Wr))/Q_(c)/Q_(r) W_(c)/W_(r) V_((r,Qr)) W_((r,Vr)) T_((c,Qr)) W_((r,Qr))V_((r,Vr)) Q_((r,Vr)) R_((c,Wr)) Q_((r,Wr)) Vanadium diBoride 0.8851.864 1.13 2.10 >T oxid. 0.326 0.475 439.337 0.648 Vanadium Carbide0.752 2.341 1.33 3.11 >T oxid. 0.26 0.32 392.1 0.49 Vanadium Nitride0.74 2.1 1.34 2.83 >T oxid. 0.289 0.353 309.9 0.513 Zirconium Carbide0.425 2.02 2.35 4.75 >T oxid. 0.301 0.21 422.05 0.299 Zirconium Nitride0.64 1.5 1.57 2.36 >T oxid. 0.405 0.423 408.1 0.518

[0068] A Thermoelastic Actuator

[0069] A hot arm/cold arm actuator is presented in FIG. 1 to illustratethe results contained in Table 3. Only the steady state solution for aquantity of heat input to the heater is analyzed. The device comprisestwo identical material layers separated by air and connected to eachother at the ends by a thermally non-conductive block. Theforce/deflection characteristics of the output mechanical power can betuned by altering the separation between the two layers. A greaterseparation increases the transverse force but decreases deflection.

[0070] Two actuators constructed from titanium and silicon are comparedusing graphed energy results in FIG. 4. Five design comparisons forTitanium are plotted according to the results contained in Table 3. Therelationship between volumes, mechanical work and heat energy areidentical to those included in Table 3. Titanium volumes are scaledusing length for Comparisons 2 and 4.

[0071] Discussion

[0072] The combination of five separate material properties is importantin assessing a material's potential for thermoelastic design andmaterials with one predominant property have been shown to notnecessarily be the best candidate. This is evident in both Table 3 forPTFE (high g) and diamond (high E). Both gold and copper have high gvalues but are hindered as good candidates by low E and high r values.Silicon is very inefficient compared to certain other materials,however, amorphous silicon dioxide is possibly the most inefficientmaterial of all.

[0073] Output mechanical work, input heat energy and actuator volume arethree essential characterizing parameters for thermoelastic design. Thedesign method described incorporates these parameters using onlymaterial properties and provides a structured approach for materialselection. The method is versatile because the approach assesses thepotential of a material using easily calculated comparison ratios. It isimportant to note that the approach is a measure of a materialspotential and must be used as a tool in conjunction with otherappropriate design criteria. For example, criteria such asforce/deflection characteristics of the output work, materialresistivity, environmental ruggedness and material availability may beimportant. The operable temperature range is assumed to be from 0degrees to the melting point on the Centigrade scale because it isindicative of the material thermal range. However, the maximum operabletemperature could be different due to oxidation of the material or otherthermal design constraints. Titanium nitride has close to the highestactuation efficiency value when melting point is used as a criteria.However, Titanium diSilicide is potentially a better candidate for usewhen oxidation temperature is used. Titanium nitride is a practicalcandidate because it is well established as a CMOS barrier material. Theoxidation temperature of TiN can be raised from 500° C. to 900° C. byalloying with aluminum. The alloyed material has a symbol (Ti,Al)N.

[0074] The actuation efficiency of a simple thermoelastic titanium beamis low compared to other actuation mechanisms (less than 1 percent). Itis theoretically possible to get a thermoelastic actuation efficiency ofabout 4.5 percent for a simple titanium nitride beam, however, thisvalue typically decreases when the material is implemented in a MEMSdevice due to associated operational losses (for example—thermalconduction into the substrate).

[0075] The invention has been described herein by way of example only.Skilled workers in this field will readily recognize many variations andmodifications which do depart from the spirit and scope of the broadinventive concept.

We claim:
 1. A method of selecting a material for use as the expansiveelement in a thermoelastic design by deriving an indicator of thematerial's potential effectiveness for that use, said method includingthe step of calculating a dimensionless constant εγ for that material inaccordance with the formula:${ɛ\quad \gamma} = \frac{E\quad \gamma^{2}T}{\rho \quad C}$

wherein E is the Young's modulus of the material; γ is the coefficientof thermal expansion; T is the maximum operating temperature, ρ is thedensity and C is the specific heat capacity and selecting the materialon the basis of ε.
 2. The method of claim 1 further including the stepof normalizing the dimensionless constant relative to that of silicon toa value ε which is achieved by deriving the value εγ for the material ofinterest at the relevant temperature value and dividing this by thevalue of ε obtained for silicon at that same temperature.
 3. The methodof claim 1 further including determining m_(ε) where $\begin{matrix}{m_{ɛ} = {\frac{ɛ}{T} = {\frac{E\quad \gamma^{2}}{\rho \quad C}\left\lbrack \frac{N\text{/}m^{2}\quad 1\text{/}{^\circ}\quad {C.^{2}}}{{kg}\text{/}m^{3}\quad {Nm}\text{/}{kg}\quad {^\circ}\quad {C.}} \right\rbrack}}} & \left( {{EQ}\quad 6} \right)\end{matrix}$

and selecting the material on the basis of both m_(ε) and ε
 4. Themethod of claim 1 further including the step of eliminating certainmaterials by requiring a pre-determined resistivity range.
 5. The methodof claim 3 further wherein the resistivity range is between 0.1 μΩm and10.0 μΩm.
 6. The method of claim 1 including selecting a material on thebasis of at least one of the following group of properties: aresistivity between 0.1 μΩm and 10.0 μΩm; chemically inert in air;chemically inert in the chosen ink; and depositable by CVD, sputteringor other thin film deposition technique.
 7. The method of claim 1including selecting a material from a group including: silicides andcarbides of titanium; borides, suicides, carbides and nitrides oftantalum, molybdenum, niobium, chromium, tungsten, vanadium, andzirconium.
 8. A method of manufacturing a thermoelastic element thatincludes at least one expansive element, the method including: selectinga material for use as the expansive element in the thermoelastic designby deriving an indicator of the material's potential effectiveness forthat use, said method including the step of calculating a dimensionlessconstant εγ for that material in accordance with the formula:${ɛ\quad \gamma} = \frac{E\quad \gamma^{2}T}{\rho \quad C}$

wherein E is the Young's modulus of the material; γ is the coefficientof thermal expansion; T is the maximum operating temperature, ρ is thedensity and C is the specific heat capacity and selecting the materialon the basis of ε, and manufacturing the thermoelastic element with theat least one expansive element formed of the selected material.
 9. Themethod of claim 8 further including the step of normalizing thedimensionless constant relative to that of silicon to a value ε which isachieved by deriving the value εγ for the material of interest at therelevant temperature value and dividing this by the value of ε obtainedfor silicon at that same temperature.
 10. The method of claim 8 furtherincluding determining m_(ε) where $\begin{matrix}{m_{ɛ} = {\frac{ɛ}{T} = {\frac{E\quad \gamma^{2}}{\rho \quad C}\left\lbrack \frac{N\text{/}m^{2}\quad 1\text{/}{^\circ}\quad {C.^{2}}}{{kg}\text{/}m^{3}\quad {Nm}\text{/}{kg}\quad {^\circ}\quad {C.}} \right\rbrack}}} & \left( {{EQ}\quad 6} \right)\end{matrix}$

and selecting the material on the basis of both m_(ε) and ε
 11. Themethod of claim 8 further including the step of eliminating certainmaterials by requiring a pre-determined resistivity range.
 12. Themethod of claim 11 further wherein the resistivity range is between 0.1μΩm and 10.0 μΩm.
 13. The method of claim 8 including selecting amaterial on the basis of at least one of the following group ofproperties: a resistivity between 0.1 μΩm and 10.0 μΩm; chemically inertin air; chemically inert in the chosen ink; and depositable by CVD,sputtering or other thin film deposition technique.
 14. The method ofclaim 8 including selecting a material from a group including: suicidesand carbides of titanium; borides, suicides, carbides and nitrides oftantalum, molybdenum, niobium, chromium, tungsten, vanadium, andzirconium.